On the Geometry of Piecewise Circular Curves

Abstract

In this article we would like to promote a class of plane curves that have a number of special and attractive properties, the piecewise circular curves, or PC curves. (We feel constrained to point out that the term has nothing to do with Personal Computers, Privy Councils, or Political Correctness.) They are nearly as easy to define as polygons: a PC curve is given by a finite sequence of circular arcs or line segments, with the endpoint of one arc coinciding with the beginning point of the next. These curves are more versatile than polygons in that they can have a well-defined tangent line at every point: a PC curve is said to be smooth if the directed tangent line at the end of one arc coincides with the directed tangent line at the beginning of the next. (In particular, in a smooth PC curve, no arc degenerates to a single point.) In the literature of descriptive geometry and more recently in computer graphics, PC curves have been used to approximate smooth curves so that the approximation is not only pointwise close, as in the case of an inscribed polygon, but also has the property that the tangent lines at the points of the smooth curve are approximated by the tangent lines of the PC curve. Given a pair of nearby points on a smooth curve together with their tangent directions, there will not in general be a single circular arc through the points with those directions at its endpoints, but there will be a family of biarcs meeting these boundary conditions, PC curves composed of two tangent circular arcs. (See [M-N] for a discussion of this construction.) EXAMPLES OF PC CURVES. PC curves arise naturally as the solutions of a number of variational problems related to isoperimetric problems. A classical problem is to find the curve of shortest length enclosing a fixed area, and the solution is a circle. If the curve is required to surround a fixed pair of points, then the curve of shortest length enclosing a given area will be either a circle or a lens formed by two arcs of circles of the same radius meeting at the two points. More generally Besicovitch has shown that a curve of fixed length surrounding a given convex polygon and enclosing the maximum area must be a PC curve with all radii of arcs equal [Be]. One such curve is the Reuleaux a three-arc PC curve enclosing an equilateral triangle, with each radius equal to the length of a side of the triangle. Such three-arc PC curves, and many far more elaborate examples can be found in the tracery of gothic windows [A]. If we require that a curve of fixed length L surround a given pair of discs of the same radius, then, for a certain range of values of L, the curve that encloses the greatest area is a smooth convex PC curve consisting of two arcs on the boundary circles of the discs and two arcs of equal radius tangent to both discs. Such four-arc convex PC curves have long been used in engineering drawing for approximating ellipses, and we call such a curve a PC ellipse [FIGURE 1]. One special PC ellipse is